Since \(I_{t,x}\)  is finite only for absolutely continuous functions, it suffices to consider sequences in \(C([0,T];A)\cap \lbrace \phi | \phi (0)=x\rbrace \) . Limits of such sequences (under either metric) are continuous and convergence is equivalent for both metrics (see, e.g., [1]}). Lower semicontinuity follows. Compactness of the level sets follows by Proposition REF  and the fact that the identity maps from \((C([0,T];A),d_C)\)  into \((D([0,T];A),d_C)\)  and \((D([0,T];A),d_D)\)  are continuous .
